A variant of Wigner's theorem in normed spaces
Let \(X\) and \(Y\) be normed spaces over \(\mathbb{F}\) and let \(U:X \rightarrow Y\) be a linear (or a conjugate linear) isometry. If a function \(f:X \rightarrow Y\) has the property \[ f(x)=\sigma(x) Ux, \quad x\in H, \] where \(\sigma\) is a phase function, i.e., \(\sigma\) takes values in modulus one scalars, then a function \(f\) is called phase equivalent to a linear (or a conjugate linear) isometry. The main result of this paper is given in the following theorem. Theorem. Let \(X\) and \(Y\) be normed spaces over \(\mathbb{F}\) and \(f:X \rightarrow Y\) a surjective mapping. Suppose that for all semi-inner products on \(X\) and \(Y\), we have \[ | [f(x),f(y)]| = |[x,y]|, \quad x,y \in X.\] The following statements hold true: \begin{itemize} \item[(i)] If dim \(X=1\), then \(f\) is phase equivalent to a linear surjective isometry; \item[(ii)] If dim \(X\geq 2\) and \(\mathbb{F} = \mathbb{R}\), then \(f\) is phase equivalent to a linear surjective isometry; \item[(iii)] If dim \(X\geq 2\) and \(\mathbb{F} =\mathbb{C}\), then \(f\) is phase equivalent to a linear or conjugate linear surjective isometry. \end{itemize}
scientific article; zbMATH DE number 7385405
scientific article